Complex Dynamics and Moduli Spaces

A central issue in science, from biology to physics, is the classification of the wide variety of shapes that a single recognized type can assume. In mathematics, the basic type might be something as concrete as a polyhedron or a loop in the plane, or as complicated as a holomorphic 1-form on an algebraic curve. In all cases, a moduli space intervenes to describe the classification. This project aims to study moduli spaces from the perspective of analysis, algebra, geometry and dynamics and by employing and integrating AI tools; to discover exotic new moduli spaces, hitherto unseen; and to illuminate the arithmetic symmetries and universal laws governing these spaces. The project also centers on mentoring, computation and communication. It aims to: (i) Mentor and train Ph.D. students to join the next generation of American mathematicians as independent researchers --- an undertaking that is absolutely essential for the nation's economic security, and for its continued leadership in scientific progress worldwide; (ii) Leverage the rapidly changing world of computation, from traditional pursuits such as computer graphics and algebraic manipulation, to the use of AI to attack new problems and enrich the research process; and (iii) Communicate the excitement of current research through meetings and lectures around the world, ranging from research-level seminars to public popularizations of science. For loops in the plane, the project proposes a universal meta-Teichmueller theory to provide a moduli space for fractal shapes arising in many apparently unrelated areas of complex analysis. For the moduli space of negatively curved 3-manifolds, it will explore bialgebraic points and formulate a new version of the Andre'-Oort conjecture in this setting - a conjecture which has already been a driving force for research in Diophantine geometry. In the case of holomorphic 1-forms, the project aims to solve a problem that has been open for decades: to give a complete classification of the rare, extremal objects known as Teichmueller curves, and to illuminate the objects themselves from multiple perspectives, ranging from topology to number theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2552971 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Curtis McMullen | Institution: Harvard University, CAMBRIDGE, MA | Award Amount: $300,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2552971 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2552971.html